FUNCTION: ToXfixScal - convert any expression to the X Schubert basis of the ring of polynomials as a free module over Sym

CALLING SEQUENCE:

ToXfixScal(expr)

ToXfixScal(expr, b)

FM[ToXfixScal](expr)

FM[ToXfixScal](expr, b)

PARAMETERS:

expr = any expression

b = any name of a known basis

SYNOPSIS:

• The ToXfixScal function converts any expression expr to the X Schubert basis of the ring of polynomials as a free module over free module over symmetric polynomials, by using scalar products instead of Monk formula when possible. This function is useful for _FMn <= 5 especially when pre-computed data tables are loaded. expr may involve some xi's, simple Schubert polynomials (X[perm], Y[code]), other terms being considered as coefficients.

• The expression expr is expanded and the result is not collected.

• One may specify by a second argument, say b, that expr is solely expressed in terms of the known basis b (x, X, Y).

• The call ToXfixScal(expr, 'X') does not affect the argument expr.

• One may add 'noexpand' just after the argument expr to choose not to expand the expression expr before treating it.

• One may collect the result by adding a third argument: this is done by ToXfixScal(expr, b, 'collect'). For instance, the next instruction ToXfixScal(expr, 'X', 'collect') may be used to collect its argument.

• Whenever there is a conflict between the function name ToXfixScal and another name used in the same session, use the long form FM['ToXfixScal'].

EXAMPLES:

> with(FM):
> FM_n(4);
 
                                      4
 
> ToXfixScal((1+q)^2*x3*x4):                     # expands the input
> ToXfixScal((1+q)^2*x3*x4,noexpand):            # does not expand (1+q)^2
> ToXfixScal((1+q)^2*x3*x4,collect):             # collects the result
> ToXfixScal((1+q)^2*x3*x4,noexpand,'x'):        # the most efficient
> ToXfixScal((1+q)^2*x3*x4,'x',collect):         # specifies a basis
> ToXfixScal((1+q)^2*x3*x4,noexpand,'x',collect):
> ToXfixScal(q^2*x3*X[3,1,4,2]*Y[0,1], collect);
 
       2              2                  2
    - q  X[1, 3, 2, 4]  X[1, 2, 3, 4] + q  s[1] X[3, 4, 1, 2]
 
       2                       2                       2
    + q  s[1] X[3, 2, 4, 1] + q  s[1] X[4, 1, 3, 2] + q  X[3, 4, 2, 1]
 
       2                  2
    + q  X[4, 2, 3, 1] - q  s[1, 1, 1] X[2, 3, 1, 4]
 
       2                             2
    - q  s[1, 1, 1] X[3, 1, 2, 4] + q  s[1, 1, 1, 1] X[1, 3, 2, 4]